Use Boolean algebras to prove the following

Question

Use Boolean algebras to prove the following: $x ∨ ((x ∨ y) ∧ (x ∨ y)) = x ∨ ((x ∨ y) ∧ x) ∨ ((x ∨ y) ∧ y))$

I'm not sure but I'm starting with right-hand-side because I think it might be a little bit easier

$x ∧ x = x$ idempotent

$x ∨ ((x ∧ x) ∨ (x ∨ y)) ∨ ((x ∧ y) ∨ (y ∧ y))$ Substitution

$x ∨ (x ∨ (x ∧ y)) ∨ ((x ∧ y) ∨ y) $

I stuck here

Answer 1

HINT

Use Idempotence: $\phi \land \phi = \phi$

And Absorption: $(\phi \lor \psi) \land \phi = \phi$

So:

$x \lor ((x \lor y) \land x) \lor ((x \lor y) \land y))=$ (Absorption x2)

$x\lor (x \lor y)=$ (Idempotence)

$x \lor ((x \lor y)\land (x \lor y))$